An approach to multiple attribute decision making based on the induced Choquet integral with fuzzy number intuitionistic fuzzy information.

Wei, Guiwu ; Lin, Rui ; Zhao, Xiaofei 等

Introduction

Multiple attribute decision making (MADM) problems are wide spread in real life decision situations. A MADM problem is to find a most desirable alternative from all feasible alternatives assessed on multiple attributes, both quantitative and qualitative (Kersuliene et al. 2010; Wachowicz 2010; Zavadskas et al. 2010). Atanassov (1986) introduced the concept of intuitionistic fuzzy set(IFS), which is a generalization of the concept of fuzzy set (Zadeh 1965). The intuitionistic fuzzy set has received more and more attention since its appearance. Xu and Yager (2006) developed some geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator and gave an application of the IFHG operator to multiple attribute group decision making with intuitionistic fuzzy information. Xu (2007a) developed the intuitionistic fuzzy weighted averaging (IFWA) operator, the intuitionistic fuzzy ordered weighted averaging (IFOWA) operator, and the intuitionistic fuzzy hybrid aggregation (IFHA) operator. Wei (2008) utilized the maximizing deviation method for intuitionistic fuzzy multiple attribute decision making with incomplete weight information. Atanassov and Gargov (1989) introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFSs) as a further generalization of that of IFSs, as well as of IVFSs. Atanassov (1994) defined some operational laws of the IVIFSs. Xu and Chen (2007) developed some arithmetic aggregation operators, such as the interval-valued intuitionistic fuzzy weighted averaging (IIFWA) operator, the interval-valued intuitionistic fuzzy ordered weighted averaging (IIFOWA) operator, and the interval-valued intuitionistic fuzzy hybrid aggregation (IIFHA) operator and gave an application of the IIFHA operator to multiple attribute group decision making with interval-valued intuitionistic fuzzy information. Xu (2007b) developed some geometric aggregation operators, such as the interval-valued intuitionistic fuzzy weighted geometric (IIFWG) operator and the interval-valued intuitionistic fuzzy geometric (IIFG) operator and gave an application of the IIFWG and IIFG operators to multiple attribute group decision making with interval-valued intuitionistic fuzzy information. Ye (2009) proposed Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment. Wei (2009) investigated the dynamic intuitionistic fuzzy multiple attribute decision making problems and proposed the dynamic intuitionistic fuzzy weighted geometric (DIFWG) operator and uncertain dynamic intuitionistic fuzzy weighted geometric (UDIFWG) operator to aggregate dynamic or uncertain dynamic intuitionistic fuzzy information. Wei (2010a) investigated the multiple attribute group decision making (MAGDM) problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of intuitionisticfuzzy numbers or interval-valued intuitionisticfuzzy numbers and proposed two new aggregation perators: induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator and induced interval-valued intuitionistic fuzzy ordered weighted geometric (I-IIFOWG) operator, and studied some desirable properties of the I-IFOWG and I-IIFOWG operators, such as commutativity, idempotency and monotonicity. An I-IFOWG and IFWG (intuitionisticfuzzy weighted geometric) operators-based approach is developed to solve the MAGDM problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of intuitionisticfuzzy numbers. Further, they extended the developed models and procedures based on I-IIFOWG and IIFWG (interval-valued intuitionisticfuzzy weighted geometric) operators to solve the MAGDM problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of interval-valued intuitionisticfuzzy numbers. Li (2010) proposed Linear programming method for MADM with interval-valued intuitionistic fuzzy sets. Xu and Xia (2011) studied the inducedgeneralized aggregation operators under intuitionisticfuzzy environments. Choquet integral and Dempster-Shafer theory of evidence are applied to aggregate inuitionistic fuzzy information and some new types of aggregation operators are developed, including the induced generalized intuitionistic fuzzy Choquet integral operators and induced generalized intuitionistic fuzzy Dempster-Shafer operators. Then they investigated their various properties and some of their special cases. Additionally, they applied the developed operators to financial decision making under intuitionisticfuzzy environments. Some extensions in interval- valued intuitionisticfuzzy situations are also pointed out.

Liu and Yuan (2007) introduced the concept of fuzzy number intuitionistic fuzzy set (FNIFS) which fundamental characteristic of the FNIFS is that the values of its membership function and non-membership function are triangular fuzzy numbers rather than exact numbers. Wang (2008a) propose the fuzzy number intuitionistic fuzzy weighted averaging (FNIFWA) operator, fuzzy number intuitionistic fuzzy ordered weighted averaging (FNIFOWA) operator and fuzzy number intuitionistic fuzzy hybrid aggregation (FNIFHA) operator. Wang (2008b) propose some aggregation operators, including fuzzy number intuitionistic fuzzy weighted geometric (FNIFWG) operator, fuzzy number intuitionistic fuzzy ordered weighted geometric (FNIFOWG) operator and fuzzy number intuitionistic fuzzy hybrid geometric (FNIFHG) operator and develop an approach to multiple attribute group decision making (MAGDM) based on the FNIFWG and the FNIFHG operators with fuzzy number intuitionistic fuzzy information. Wei et al. (2010a) developed the induced fuzzy number intuitionistic fuzzy ordered weighted geometric (I-FIFOWG) operator and studied some desirable properties of the I-FIFOWG operators, such as commutativity, idempotency and monotonicity. An I-FIFOWG and FIFWG (fuzzy number intuitionistic fuzzy weighted geometric) operators-based approach is developed to solve the MAGDM under the fuzzy number intuitionistic fuzzy environment. Furthermore, they proposed the induced fuzzy number intuitionistic fuzzy ordered weighted averaging (I-FIFOWA) operator.

All of the existing fuzzy number intuitionistic fuzzy aggregation operators only consider situations where all the elements in the fuzzy number intuitionistic fuzzy set are independent. However, in many practical situations, the elements in the fuzzy number intuitionistic fuzzy set are usually correlative. Therefore, we need to find some new ways to deal with these situations in which the decision data in question are correlative. The Choquet integral is a very useful way of measuring the expected utility of an uncertain event, and can be utilized to depict the correlations of the decision data under consideration. Motivated by the correlation properties of the Choquet integral, in this paper we propose some fuzzy number intuitionistic fuzzy aggregation operators, whose prominent characteristic is that they can not only consider the importance of the elements or their ordered positions, but also reflect the correlations of the elements or their ordered positions. To do so, the remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to fuzzy number intuitionistic fuzzy sets and some operational laws of fuzzy number intuitionistic fuzzy numbers. In section 2 we have developed two induced fuzzy number intuitionistic fuzzy Choquet integral aggregation operators: induced fuzzy number intuitionistic fuzzy choquet ordered averaging (IFNIFCOA) operator and induced fuzzy number intuitionistic fuzzy choquet ordered geometric (IFNIFCOG) operator. In section 3, we have developed an approach to multiple attribute decision making based on IFNIFCOA operator and IFNIFCOG operator with fuzzy number intuitionistic fuzzy information. In section 4, an illustrative example is pointed out. In the last Section, we conclude the paper and give some remarks.

1. Preliminaries

Atanassov (1986) extended the fuzzy set to the IFS, shown as follows.

Definition 1. Given a fixed setX = {[x.sub.1], [x.sub.2] ,---,[x.sub.n] } , an intuitionistic fuzzy set (IFS) is defined as (Atanassov 1986):

A = (<[x.sub.i], [t.sub.A] ([x.sub.i]), [f.sub.A] ([x.sub.i])>/[x.sub.i] [member of] X), (1)

which assigns to each elements [x.sub.i] a membership degree [t.sub.A] ([x.sub.i]) and a non-membership degree [f.sub.A] ([x.sub.i]) under the condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Atanassov and Gargov (1989) further introduced the interval-valued intuitionistic fuzzy set (IVIFS), which is a generalization of the IFS. The fundamental characteristic of the IVIFS is that the values of its membership function and non-membership function are intervals rather than exact numbers.

Definition 2. Given a fixed set X = {[x.sub.1], [x.sub.2],--,[x.sub.n]}, then an IVIFSs [??] over X is an object having the form:

[??] = {<[x.sub.i], [[??].sub.A] ([x.sub.i]), [??].sub.A] ([x.sub.i])>| [x.sub.i] [member of] X}, (2)

where [[??].sub.A] ([x.sub.i]) e [0,l] and [[??].sub.A] ([x.sub.i]) [subset] [0,l] are interval numbers, and 0 [less than or equal to] [[??].sub.A] ([x.sub.i]) + sup [[??].sub.A] ([x.sub.i]) [less than or equal to][less than or equal to] l, [for all] [x.sub.i] [member of] X.

Liu and Yuan (2007) introduced the concept of fuzzy number intuitionistic fuzzy set (FNIFS) which fundamental characteristic of the FNIFS is that the values of its membership function and non-membership function are triangular fuzzy numbers.

Definition 3. Given a fixed set X = {[x.sub.l], [x.sub.2],-, [x.sub.n]}, an FNIFS [??] over X is an object having the form:

[??] = {<[x.sub.i], [[??].sub.A] ([x.sub.i]), [[??].sub.A] ([x.sub.i]>}/[x.sub.i] [member of] X}, (3)

where [[??].sub.A] (xt) e [0,l] and fA (xi) e [0,l] are triangular fuzzy numbers, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For convenience, let [[??].sub.A] ([x.sub.i]) = (a([x.sub.i]), b([x.sub.i]), c([x.sub.i])), [[??].sub.A](x) = (l([x.sub.i]), m([x.sub.i]), p([x.sub.i])) and we call [??] ([x.sub.i]) an fuzzy number intuitionistic fuzzy value (FNIFV).

Definition 4. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 5. Let [??]([x.sub.i]) = <(a([x.sub.i]), (b([x.sub.i]), (c([x.sub.i])), (l([x.sub.i]), m([x.sub.i]), p([x.sub.i]))> be a FNIFV, a score function S of a FNIFV [??]([x.sub.i]) can be represented as follows (Wang 2008a):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Definition 6. Let [??]([x.sub.i]), <(a([x.sub.i]), (b([x.sub.i])), (c([x.sub.i]), (b([x.sub.i]) l([x.sub.i]), p([x.sub.i]) be a FNIFV, an accuracy function H of a FNIFV [??]([x.sub.i]) can be represented as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

to evaluate the degree of accuracy of the FNIFV: [??] ([x.sub.i])) = <(a([x.sub.i]), (b([x.sub.i])), (c([x.sub.i]), (l([x.sub.i]) m([x.sub.i]), p([x.sub.i]))>

where H ([??] ([x.sub.i])) [member of] [0,l]. The larger the value of H ([??] ([x.sub.i])) , the more the degree of accuracy of the FNIFV [??] ([x.sub.i]) is.

Based on the score function S and the accuracy function H, in the following, we shall give an order relation between two fuzzy number intuitionistic fuzzy values, which is defined as follows:

Definition 7. Let [??] ([x.sub.i]) = <(a ([x.sub.i]), b([x.sub.i]), c([x.sub.i])), (l([x.sub.i]), m([x.sub.i]), p([x.sub.i]))> and

[??] ([x.sub.j]) = <(a ([x.sub.j]), b([x.sub.j]), c([x.sub.j])), (l([x.sub.j]), m([x.sub.j]), p([x.sub.j]))> be two FNIFVs, S ([??] ([x.sub.i])) and s ([??]([x.sub.j])) be the scores of [??] ([x.sub.i]) and [??] ([x.sub.j]) , respectively, and let H ([??] ([x.sub.i])) and H ([??] ([x.sub.j])) be the accuracy degrees of [??] ([x.sub.i]) and [??] ([x.sub.j]), respectively, then if s ([??] ([x.sub.i])) < s ([??] ([x.sub.j])), then [??] ([x.sub.i]) is smaller than [??] ([x.sub.j]), denoted by [??] ([x.sub.i]) < [??] ([x.sub.j]); if s ([??] ([x.sub.i])) = s ([??] ([x.sub.j])), then

(1) if H ([??] ([x.sub.i])) = H ([??] ([x.sub.j])), then [??] ([x.sub.i]) and [??] ([x.sub.j] represent the same information, denoted by [??] ([x.sub.i]) = [??] ([x.sub.j]);

(2) if H ([??] ([x.sub.i])) < H ([??] ([x.sub.j])), [??] ([x.sub.i]) is smaller than [??] ([x.sub.j]), denoted by [??] ([x.sub.i]) < [??] ([x.sub.j]).

However, the above aggregation operators with fuzzy number intuitionistic fuzzy information is based on the assumption that the attribute of decision makers are independent, which is characterized by an independence axiom (Keeney, Raiffa 1976; Wakker 1999), that is, these operators are based on the implicit assumption that attributes of decision makers are independent of one another; their effects are viewed as additive. For real decision making problems, there is always some degree of inter-dependent characteristics between attributes. Usually, there is interaction among attributes of decision makers. However, this assumption is too strong to match decision behaviors in the real world. The independence axiom generally can't be satisfied. Thus, it is necessary to consider this issue.

Let [mu]([x.sub.i]) (i = l, 2, ..., n) be the weight of the elements [x.sub.i] [member of] X (i = 1, 2, ..., n) , where [mu] is a fuzzy measure, defined as follows:

Definition 8 (Wang, Klir 1992). A fuzzy measure [mu] on the set X is a set function [mu]: [theta] (x) [right arrow] [0,l] satisfying the following axioms:

(1) [mu]([phi]) = 0 , [mu](X) = 1;

(2) A [subset or equal to] B implies [mu](A) [less than or equal to] [mu] (B), for all A,B [subset or equal to] X;

(3) [mu](A [union] B) = [mu](A) + [mu](B) + [rho][mu](A)[mu](B), for all A,B [subset or equal to] and A [intersection] B = [phi], where [rho] [member of] (-l, [infinity]).

Especially, if [rho] = 0, then the condition (3) reduces to the axiom of additive measure:

[mu](A [union] B) = [mu](A) + [mu](B), for all A,B [subset or equal to] X and A [intersection] B = [phi].

If all the elements in X are independent, and we have

[mu](A)= [summation over [x.sub.i] [member of] A]({[x.sub.i]}), for all A [subset or equal to] X.

Definition 9 (Grabisch et al., 2000). Let f be a positive real-valued function on X, and [mu] be a fuzzy measure on X. The discrete Choquet integral of f with respective to [mu] is defined by:

C[mu] (f) = [n.summation over (i=1)] [f.sub.[sigma](i)] [[mu] [A.sub.[sigma](i)] - [mu](A[[sigma](i-1)]),

where ([sigma](1), [sigma](2), ..., [sigma](n)) is a permutation of (l,2,n) , such that [f.sub.[sigma](i-1)] [greater than or equal to] [f.sub.[sigma](i)] for all j = 2, ..., n, [A.sub.[sigma](k)] = {[x.sub.[sigma]|j [less than or equal to] k}, for k [greater than or equal to] l, and [A.sub.[sigma](0)] = [phi].

It is seen that the discrete Choquet integral is a linear expression up to a reordering of the elements.

Definition 10 (Grabisch et al. 2000). Let f be a positive real-valued function on X and m be a fuzzy measure on X. The induced Choquet ordered averaging operator of dimension n is a function I-COA: ([R.sup.+] x [R.sup.+]) [right arrow] [R.sup.+], which is defined to aggregate the set of second argument of a list of 2-tuples (([u.sub.1],[f.sub.1]),([u.sub.2],[f.sub.2]), ..., <[u.sub.n], [f.sub.n]}) according to the following expression:

[I-COA.sub.m] (<[u.sub.1], [f.sub.1]>,<[u.sub.2],[f.sub.2>,..., <[u.sub.n], [f.sub.n]>)) =

[n.summation over (j=1)] [f.sub.[sigma](j)][m([A.sub.[sigma](j)]- m([A.sub.[sigma](j-1)], (6)

where ([sigma](1), [sigma](2), ..., [sigma](n)) is a permutation of (1, 2, ..., n) , such that [u.sub.[sigma](i-1)] [greater than or equal to] [u.sub.[sigma](i)] for all j = 2, ..., n, i.e., <[u.sub.[sigma](j)], [f.sub.[sigma](j)] is the 2-tuple with [u.sub.[sigma](j)] the jth largest value in the set ([u.sub.1], [u.sub.2], ..., [u.sub.n]), [A.sub.[sigma](k)]) = {[x.sub.[sigma](j)])|j [less than or equal to] k} , for k [greater than or equal to] 1, and A0(0) = [A.sub.[sigma](0)].

2. Some induced aggregating operators based on the Choquet integral with fuzzy number intuitionistic fuzzy information

Wang (2008a) propose the fuzzy number intuitionistic fuzzy weighted averaging (FNIFWA) operator and fuzzy number intuitionistic fuzzy ordered weighted averaging (FNIFOWA) operator.

Definition 11. Let [??]([x.sub.i]) = <(a([x.sub.i]),b([x.sub.i]),c([x.sub.i])),(l([x.sub.i]),m([x.sub.i]),p([x.sub.i] ))> (i = l, 2, ..., n) be a collection of FNIFVs, and let FNIFWA: [Q.sup.n] [right arrow] Q, if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [omega] = [([[omega].sub.1], [[omega].sub.1], ..., [[omega].sub.n]).sup.T] be the weight vector of [??] ([x.sub.i])(i = 1, 2, ..., n), and [[omega].sub.j] > 0, [n.summation over (i=1)] [[omega].sub.i] = 1, then FNIFWA is called the fuzzy number intuitionistic fuzzy weighted averaging (FNIFWA) operator.

Example 1. Assume [omega] = <(0.2,0.l,0.3,0.4), [[??].sub.1] =((0.l,0.2,0.3),(0.5,0.6,0.7)>, [[??].sub.2] = <(0.4,0.5,0.6), (0.3,0.3,0.4)>, [[??].sub.3] = <(0.4,0.4,0.5), (0.4,0.4,0.5)>, and [[??].sub.4] = <(0.3,0.4,0.5), (0.3,0.4,0.4)>, then

[FNIFWA.sub.[omega]] ([[??].sub.1], [[??].sub.2], [[??].sub.3], [[??].sub.4]) =

<(1 - [(1 - 0.l).sup.0.2] x [(1- 0.4).sup.0.1] x [(1 - 0.4).sup.0.3] x [(1 - 0.3).sup.0.4] J

1 - [(1 - 0.2).sup.0.2] x [(1 - 0.5).sup.0.1] x [(1 - 0.4).sup.0.3] x [(1 - 0.4).sup.0.4] J

1 - [(1 - 0.3).sup.0.2] x [(1 - 0.6).sup.0.1] x [(1 - 0.5).sup.0.3] x [(1 - 0.5).sup.0.4]),

([0.5.sup.0.2] x [0.3.sup.0.1] x [0.4.sup.0.3] x [0.3.sup.0.4],

[0.6.sup.0.2] x [0.3.sup.0.1] x [0.4.sup.0.3] x [0.4.sup.0.4],

[0.7.sup.0.2] x [0.4.sup.0.1] x [0.5.sup.0.3] x [0.4.sup.0.4])) =

<(0.308,0.376,0.477), (0.362,0.42l, 0.478)>.

Definition 12. Let [??]([x.sub.i]) = <(a ([x.sub.i]), b ([x.sub.i]), c ([x.sub.i])), (l ([x.sub.i]), m ([x.sub.i]), p ([x.sub.i]))> (i = 1, 2, ..., n) be a collection of FNIFVs, An fuzzy number intuitionistic fuzzy ordered weighted averaging (FNIFOWA) operator of dimension n is a mapping FNIFOWA: [Q.sup.n] [right arrow] Q, that has an associated weight vector w = [([w.sub.1], [w.sub.2], ..., [w.sub.n]).sup.T] such that [w.sub.j] > 0 and [n.summation over (j=1) [w.sub.j] = 1. Furthermore:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where ([sigma](1), [sigma](2), ..., [sigma](n)) is a permutation of (1, 2, ..., n), such that

[??]([x.sub.[sigma](i-1)]) [greater than or equal to] [??]([x.sub.[sigma](i)]) for all i = 2, ..., n.

Example 2. Let [[??].sub.1] = <(0.3,0.4,0.5), (0.2,0.3,0.4)), [[??].sub.2] = <( 0.3,0.3,0.3), (0.4,0.5,0.6)>, [[??].sub.3] = <(0.5,0.5,0.5), (0.3,0.3,0.3)>, and [[??].sub.4] = <( 0.1,0.2,0.2), (0.6,0.7,0.8)> be four FNIFVs, by (4), we calculate the scores of [[??].sub.j (j = 1,2,3,4):

s ([[??].sub.1]) = 0.1, s ([[??].sub.2]) = -0.2, s ([[??].sub.1]) = 0.2, s ([[??].sub.1]) = -0.53.

Since

S([[??].sub.3]) > S([[??].sub.1]) > S([[??].sub.2]) > S([[??].sub.4]),

thus

[[??].sub.[sigma](1)] = <(0.5,0.5,0.5), (0.3,0.3,0.3)>, [[??].sub.[sigma](2)] = <(0.3,0.4,0.5), (0.2,0.3,0.4)>, [[??].sub.[sigma](3)]= <(0.3,0.3,0.3), (0.4,0.5,0.6)>, [[??].sub.[sigma](4)] = <( 0.1,0.2,0.2), ( 0.6,0.7,0.8)>.

Suppose that =(0.2,0.3,0.4,0.l)is the weighting vector of the FNIFOWA operator. Then, by (7), it follows that:

[FNIFOWA.sub.w] ([[??].sub.1], [[??].sub.2], [[??].sub.3], [[??].sub.4]) =

<(1 - [(1 - 0.5).sup.0.2] x [(1 - 0.3).sup.0.3] x [(1 - 0.3).sup.0.4] x [(1 - 0.1).sup.0.1] J =

1 - [(1 - 0.5).sup.0.2] x [(1 - 0.4).sup.0.3] x [(1 - 0.3).sup.0.4] x [(1 - 0.2).sup.0.1] J

1 - [(1 - 0.5).sup.0.2] x [(1 - 0.5).sup.0.3] x [(1 - 0.3).sup.0.4] x [(1 - 0.2).sup.0.1]),

([0.3.sup.0.2] x [0.2.sup.0.3] x [0.4.sup.0.4] x [0.6.sup.0.1],

[0.3.sup.0.2] x [0.3.sup.0.3] x [0.5.sup.0.4] x [0.7.sup.0.1],

[0.3.sup.0.2] x [0.4.sup.0.3] x [0.6.sup.0.4] x [0.8.sup.0.1])> =

<(0.329,0.367,0.400), (0.319,0.401,0.476)>.

Based on Definition 10, in what follows, we shall develop the induced fuzzy number intuitionistic fuzzy choquet ordered averaging (IFNIFCOA) operator based on the well-known Choquet integral (Choquet 1953).

Definition 13. Let X ([x.sub.1], [x.sub.2], ..., [x.sub.n]) be a finite set, and [mu] be a fuzzy measure on X, and [??]([x.sub.i]) = ((a ([x.sub.i]), b ([x.sub.i]), c ([x.sub.i])), (l ([x.sub.i]), m ([x.sub.i]), p ([x.sub.i])>) (i = l, 2, ..., n) be a collection of FNIFVs on X, and [mu] be a fuzzy measure on X. An induced fuzzy number intuitionistic fuzzy choquet ordered averaging (IFNIFCOA) operator of dimension n is a function IFNIFCOA: [Q.sup.n] [right arrow] Q, which is defined to aggregate the set of second arguments of a collection of 2-tuples (<[u.sub.1], [??] ([x.sub.i])>,( [u.sub.2], [??] ([x.sub.2])>,([u.sub.n], [??] ([x.sub.n])>) according to the following expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [u.sub.i] in 2-tuple <[x.sub.i], [??] ([x.sub.i])> is referred to as the order-inducing variable and [??] ([x.sub.i]) as the argument variable, ([sigma](1), [sigma](2), ..., [sigma](n)) is a permutation of (1, 2, ..., n), such that

[u.sub.[sigma](j-l)] [greater than or equal to] [u.sub.[sigma](j)] for all j = 2, ..., n, [A.sub.(i)] = {[x.sub.(1)], [x.sub.(2)], ..., [x.sub.(i)]} when i [x.sub.(1)] 1 and [A.sub.[sigma](0)] = [phi].

With the operation of fuzzy number intuitionistic fuzzy numbers, the IFNIFCOA operator can be transformed into the following from by induction on n:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

whose aggregated value is also a fuzzy number intuitionistic fuzzy number. Especially, if [mu]({[x.sub.[sigma](j)]}) = [mu]([A.sub.[sigma](j)]) - [mu] ([A.sub.[sigma](j-1)], i = 1, 2, ..., n, then IFNIFCOA operator reduce to IFNIF WA operator. If [mu](A) = [summation over [x.sub.j] [member of] A] [mu]({[x.sub.j]}), for all A [subset or equal to] X, where [absolute value of A] is the number of the elements in the set A, then [w.sub.j] = [mu]([A.sub.[sigma](j)]) - [mu]( [A.sub.[sigma](j-1)]), i = 1, 2,..., n, where w = [([w.sub.1], [w.sub.2], ..., [w.sub.n]).sup.T] , [w.sub.j] [greater than or equal to] 0, i = 1, 2, ..., n, and [n.summation over (j=1)] [w.sub.j] = 1, then, IFNIFCOA operator reduce to FNIFOWA operator. y=l

It's easy to prove that the IFNIFCOA operator has the following properties.

Theorem 1 (Commutativity).

[IFNIFCOA.sub.[mu]] ([??]([x.sub.1]), [??]([x.sub.2]), ..., [??]([x.sub.n])) = [IFNIFCOA.sub.[mu]] ([??]'([x.sub.1]), [??]'([x.sub.2]), ..., [??]'([x.sub.n])), where ([??]'([x.sub.1]), [??]'([x.sub.2]), ..., [??]'([x.sub.n])) is any permutation of ([??]'([x.sub.1]), [??]'([x.sub.2]), ..., [??]'([x.sub.n])).

Theorem 2 (Idempotency). If [??]([x.sub.j]) = [??](x) for all j, then:

[IFNIFCOA.sub.[mu]] ([??]([x.sub.1]), [??]([x.sub.2], ..., [??]([x.sub.1])) = [??](x)

Theorem 3 (Monotonicity). If [??]([x.sub.j]) [less than or equal to] [??]' ([x.sub.j]) for all j, then:

[IFNIFCOA.sub.[mu]] ([??]([x.sub.1]), [??]([x.sub.2]), ..., [??]([x.sub.n])) [less than or equal to]

[IFNIFCOA.sub.[mu]] ([??]'([x.sub.1]), [??]'([x.sub.2]), ..., [??]'([x.sub.n]))

Example 3. Assume we have four FNIFOWA pairs <[u.sub.j, [??]([x.sub.j])> given:

<[u.sub.1],[??]([x.sub.1])> = <(l2,((0.3,0.4,0.5),(0.1,0.2,0.3))>, <[u.sub.2],[??]([x.sub.2])> = <l5, (( 0.4,0.5,0.5), ( 0.1,0.1,0.2 ))>, <[u.sub.3],[??]([x.sub.3])> = <l0, (( 0.2,0.3,0.3), ( 0.3,0.4,0.5))>, <[u.sub.4],[??]([x.sub.4])> = <8,((0.2,0.3,0.4),(0.4,0.5,0.6))>.

That we desire to aggregate using the weighting vector w = (0.2,0.4,0.1,0.3). Performing the ordering the FNIFOWA pairs with respect to the first component, we get:

<[u.sub.[sigma](1),[??]([x.sub.[sigma](1)])> = <15,((0.4,0.5,0.5),(0.1,0.1,0.2))>, <[u.sub.[sigma](2),[??]([x.sub.[sigma](2)])> = <12,((0.3,0.4,0.5),(0.i,0.2,0.3))>, <[u.sub.[sigma](3),[??]([x.sub.[sigma](3)])> = <10,((0.2,0.3,0.3),(0.3,0.4,0.5))> <[u.sub.[sigma](4),[??]([x.sub.[sigma](4)])> = <8,((0.2,0.3,0.4),(0.4,0.5,0.6))>.

This ordering includes the ordered fuzzy number intuitionistic fuzzy numbers:

{??]([x.sub.[sigma](1)) = <(0.4,0.5,0.5), (0.1,0.1,0.2)), {??]([x.sub.[sigma](2)) = <(0.3,0.4,0.5), (0.1,0.2,0.3)), {??]([x.sub.[sigma](3)) = <(0.2,0.3,0.3), (0.3,0.4,0.5)), {??]([x.sub.[sigma](4)) = <(0.2,0.3,0.4), (0.4,0.5,0.6)).

Suppose the fuzzy measure of attribute of [[??].sub.i](i = 1,2,3,4) and attribute sets of [[??].sub.i] (i = 1,2,3,4) as follows:

[mu]([phi]) = 0, [mu]([[??].sub.1]) = 0.40, [mu]([[??].sub.2]) = 0.25, [mu]([[??].sub.3]) = 0.38, [mu]([[??].sub.4]) = 0.25, [mu]([[??].sub.1], [[??].sub.2]) = 0.56, [mu]([[??].sub.1], [[??].sub.3]) = 0.65, [mu]([[??].sub.1], [[??].sub.4]) = 0.50, [mu]([[??].sub.2], [[??].sub.3]) = 0.45, [mu]([[??].sub.2], [[??].sub.4]) = 0.39, [mu]([[??].sub.3], [[??].sub.4]) = 0.40, [mu]([[??].sub.1], [[??].sub.2], [[??].sub.3]) = 0.80, [mu]([[??].sub.1], [[??].sub.2], [[??].sub.4]) = 0.75, [mu]([[??].sub.1], [[??].sub.3], [[??].sub.4]) = 0.72, [mu]([[??].sub.2], [[??].sub.3], [[??].sub.4]) = 0.60, [mu]([[??].sub.1], [[??].sub.2], [[??].sub.3], [[??].sub.4])) = 1.00.

Then, by (10), it follows that:

[IFNIFCOA.sub.[mu]] (<[u.sub.1], [??]([x.sub.1]>), <[u.sub.2], [??]([x.sub.2]>, <[u.sub.3], [??]([x.sub.3]>, <[u.sub.4], [??]([x.sub.4]>) = <(1 - [(1 - 0.4).sup.0.25] x [(1 - 0.3).sup.0.56-0.25] x [(1 - 0.2).sup.0.80- 0.56] x [(1 - 0.1).sup.1-0.80] J 1 - [(1 - 0.5).sup.0.25] x [(1 - 0.4).sup.0.56-0.25] x [(1 - 0.3).sup.0.80-0.56] x [(1 - 0.2).sup.1-0.80] J 1 - [(1 - 0.5).sup.0.25] x [(1 - 0.5).sup.0.56-0.25] x [(1 - 0.3).sup.0.80-0.56] x [(1 - 0.3).sup.1-0.80]),

([0.1.sup.0.25] x [0.1.sup.0.56-0.25] x [0.3.sup.0.80-0.56] x [0.3.sup.1-0.80], [0.l.sup.0.25] x [0.2.sup.0.56-0.25] x [0.4.sup.0.80-0.56] x [0.4.sup.l-0.80], [0.2.sup.0.25] x [0.3.sup.0.56-0.25] x [0.5.sup.0.80-0.56] x [0.5.sup.1-0.80])> = <(0.269,0.370,0.420), (0.162,0.228,0.339)>.

Wang (2008b) propose the fuzzy number intuitionistic fuzzy weighted geometric (FNIFWG) operator and fuzzy number intuitionistic fuzzy ordered weighted geometric (FNIFOWG) operator.

Definition 14. Let [??]([x.sub.i]) = <(a ([x.sub.i]), b ([x.sub.i]), c ([x.sub.i])), (l ([x.sub.i]), m ([x.sub.i]), p ([x.sub.i]))> = l, 2, ..., n) be a collection of FNIFVs, and let FNIFWG: [Q.sup.n] [right arrow] Q if:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where [omega] = [([[omega].sub.1], [[omega].sub.1], ..., [[omega].sub.n]).sup.T] be the weight vector of [??]([x.sub.i])(i = 1, 2, ..., n), and [[omega].sub.i] > 0, [n.summation over (i=1)] [[omega].sub.1] = 1, then FIFWG is called the fuzzy number intuitionistic fuzzy weighted geometric (FIFWG) operator.

Example 4. Assume [omega] = (0.2,0.1,0.3,0.4), [[??].sub.1] = <(0.1,0.2,0.3), (0.5,0.6,0.7)>, [[??].sub.2] = <(0.4,0.5,0.6), (0.3,0.3,0.4)>, [[??].sub.3] = <(0.4,0.4,0.5), (0.4,0.4,0.5)>, and [[??].sub.4] = <(0.3,0.4,0.5), (0.3,0.4,0.4)>, then: [FNIFWG.sub.[omega]] ([[??].sub.1], [[??].sub.2], [[??].sub.3], [[??].sub.4]) = <[[0.1.sup.0.2] x [0.4.sup.0.1] x [0.4.sup.0.3] x [0.3.sup.0.4], [0.2.sup.0.2] x [0.5.sup.0.1] x [0.4.sup.0.3] x [0.4.sup.0.4], [0.3.sup.0. x [0.6.sup.0.1] x [0.5.sup.0.3] x [0.5.sup.0.4]), (1 - [(1 - 0.5).sup.0.2] x [(1 - 0.3).sup.0.1] x [(1 - 0.4).sup.0.3] x [(1 - 0.3).sup.0.4] J 1 - [(1 - 0.6).sup.0.2] x [(1 - 0.3).sup.0.1] x [(1 - 0.4).sup.0.3] x [(1 - 0.4).sup.0.4] J 1 - [(1 - 0.7).sup.0.2] x [(1 - 0.4).sup.0.1] x [(1 - 0.5).sup.0.3] x [(1 - 0.4).sup.0.4])> = <(0.270,0.356,0.460), (0.625,0.562,0.495)>.

Definition 15. Let [??]([x.sub.i]) = ((a ([x.sub.i]), b ([x.sub.i]), c ([x.sub.i])), (l ([x.sub.i]), m ([x.sub.i]), p ([x.sub.i]))) (i = l, 2, ..., n) be a collection of FNIFVs, An fuzzy number intuitionistic fuzzy ordered weighted geometric (FNIFOWG) operator of dimension n is a mapping FNIFOWG: [Q.sup.n] [right arrow] Q, that has an associated weight vector w = [([w.sub.1], [w.sub.2], ..., [w.sub.n]).sup.T] such that [w.sub.i] > 0 and [n.summation over (i=1)] [w.sub.i]= 1. Furthermore: '=i

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where([sigma](l), [sigma](2), ..., [sigma](n)) is a permutation of (1, 2, ..., n), such that [??]([x.sub.[sigma](i-1)]) [greater than or equal to] [??]([x.sub.[sigma](i)]) for all i = 2, ..., n.

Example 5. Let [[??].sub.1] = <(0.3,0.4,0.5), (0.2,0.3,0.4)>, [[??].sub.2] = <( 0.3,0.3,0.3), (0.4,0.5,0.6)>, [[??].sub.3] = <(0.5,0.5,0.5), (0.3,0.3,0.3)>, and [[??].sub.4] = <( 0.1,0.2,0.2), (0.6,0.7,0.8)> be four FNIFVs, by (4), we calculate the scores of [[??].sub.j] (j = l,2,3,4):

S ([[??].sub.1]) = 0.1, S ([[??].sub.2]) = -0.2, S ([[??].sub.3]) = 0.2, S ([[??].sub.4]) = -0.53.

Since

S ([[??].sub.3]) > S ([[??].sub.1]) > S ([[??].sub.2]) > S ([[??].sub.4]), thus

[[??].sub.[sigma](1)] = <(0.5,0.5,0.5), (0.3,0.3,0.3)>, [[??].sub.[sigma](2)] = <(0.3,0.4,0.5), (02,0.3,0.4)>, [[??].sub.[sigma](3)] = <(0.3,0.3,0.3), (0.4,0.5,0.6)), [[??].sub.[sigma](4)] <(0.1,0.2,0.2), (0.6,0.7,0.8)>.

Suppose that w = (0.2,0.3,0.4,0.l)is the weighting vector of the FIFOWG operator. Then, by (11), it follows that:

[FNIFOWG.sub.[omega]] ([a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4])=

<([0.5.sup.0.2] x [0.3.sup.0.3] x [0.3.sup.0.4] x [0.1.sup.0.1], [0.5.sup.0.2] x [0.4.sup.0.3] x [0.3.sup.0.4] x [0.2.sup.0.1], [0.5.sup.0.2] x [0.5.sup.0.3] x [0.3.sup.0.4] x [0.2.sup.0.1), (1 - [(1 - 0.3).sup.0.2] x [(1 - 0.2).sup.0.3] x [(1 - 0.4).sup.0.4] x [(1 - 0.6).sup.0.1] J 1 - [(1 - 0.3).sup.0.2] x [(1 - 0.3).sup.0.3] x [(1 - 0.5).sup.0.4] x [(1 - 0.7).sup.0.1] J 1 - [(1 - 0.3).sup.0.2] x [(1 - 0.4).sup.0.3] x [(1 - 0.6).sup.0.4] x [(1 - 0.8).sup.0.1)> =

<(0.214,0.291,0.297), (0.541,0.451,0.353)>.

In the following, we shall develop the induced fuzzy number intuitionistic fuzzy choquet ordered geometric (IFNIFCOG) operator based on the well-known Choquet integral (Choquet 1953).

Definition 16. Let X ([x.sub.1], [x.sub.2], ..., [x.sub.n]) be a finite set, and [mu] be a fuzzy measure on X, and [??]([x.sub.i]) = <(a ([x.sub.i]), b ([x.sub.i]), c ([x.sub.i])), (l ([x.sub.i]), m ([x.sub.i]), p ([x.sub.i]))) (i = i, 2, ..., n) be a collection of FNIFVs on X, and [mu] be a fuzzy measure on X. An induced fuzzy number intuitionistic fuzzy choquet ordered geometric (IFNIFCOG) operator of dimension n is a function IFNIFCOG: [Q.sup.n] [right arrow] Q, which is defined to aggregate the set of second arguments of a collection of 2-tuples (<[u.sub.i], [??]([x.sub.1])>, ([u.sub.2], [??]([x.sub.2])>, ..., <[u.sub.n], [??]([x.sub.1])>) according to the following expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where [u.sub.i] in 2-tuple <[u.sub.i], [??]([x.sub.i])> is referred to as the order-inducing variable and [??]([x.sub.i]) as the argument variable, ([sigma] (1), [sigma] (2), ..., [sigma] (n)) is a permutation of (l, 2, ..., n), such that [u.sub.[sigma](j-1) [greater than or equal to] [u.sub.[sigma](j) for all j = 2, ..., n, [A.sub.(i)] = {[x.sub.(1)], [x.sub.(2)], ..., [x.sub.(i)]} when i [greater than or equal to] 1 and [A.sub.[sigma](0)] = [phi].

With the operation of fuzzy number intuitionistic fuzzy numbers, the IFNIFCOG operator can be transformed into the following from by induction on n:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

whose aggregated value is also a fuzzy number intuitionistic fuzzy number.

Especially, if [mu]({[z.sub.[sigma](j)]}) = [mu] [A.sub.[sigma](j)] - [mu] [z.sub.[sigma](j-1)]1)), i = 1, 2,..., n, then IFNIFCOG operator reduce to IFNIFWG operator. If [mu](A) = [summation over [x.sub.j] [member of] A] [mu]({[x.sub.j]}), for all A [subset or equal to] X, where [absolute value of A] is the number of the elements in the set A, then [w.sub.j] = [mu] ([A.sub.[sigma](j)]) - [mu]([A.sub.[sigma](j-1)]), i = 1, 2, ..., n, where w = [([w.sub.1], [w.sub.2], ..., [w.sub.n]).sup.T], [w.sub.j] greater than or equal to] 0, i = 1, 2, ..., n, and [n.summation over (j=1)] [w.sub.j] = 1, then, IFNIFCOG operator reduce to FNIFOWG operator.

It's easy to prove that the IFNIFCOG operator has the following properties.

Theorem 4 (Commutativity).

[IFNIFCOG.sub.[mu]] ([??]([x.sub.1), [??]([x.sub.2), ..., [??]([x.sub.n)) =

[IFNIFCOG.sub.[mu]] ([??]'([x.sub.1), [??]'([x.sub.2), ..., [??]'([x.sub.n)),

where [IFNIFCOG.sub.[mu]] ([??]'([x.sub.1), [??]'([x.sub.2), ..., [??]'([x.sub.n)) is any permutation of [IFNIFCOG.sub.[mu]] ([??]([x.sub.1), [??]([x.sub.2), ..., [??]([x.sub.n)).

Theorem 5 (Idempotency). If [??]([x.sub.j]) = [??](x) for all j, then:

[IFNIFCOG.sub.[mu]] ([??]([x.sub.1]), [??]([x.sub.2]), ..., [??]([x.sub.n])) = [??](x).

Theorem 6 (Monotonicity). If a ([x.sub.j]) [less than or equal to] [??]' (x.sub.j]) for all y, then:

[IFNIFCOG.sub.[mu]] ([??]([x.sub.1]), [??]([x.sub.2]), ..., [??]([x.sub.n])) [less than or equal to]

[IFNIFCOG.sub.[mu]] ([??]'([x.sub.1]), [??]'([x.sub.2]), ..., [??]'([x.sub.n])).

Example 6. Assume we have four FNIFOWG pairs <[u.sub.j], [??]([x.sub.j])> given:

<[u.sub.1], [??]([x.sub.1])> = <12,((0.3,0.4,0.5),(0.1,0.2,0.3))>,

<[u.sub.2], [??]([x.sub.2])> = <15, ((0.4,0.5,0.5), (0.1,0.1,0.2 )>,

<[u.sub.3], [??]([x.sub.3])> = <l0, ((0.2,0.3,0.3), (0.3,0.4,0.5))>,

<[u.sub.4], [??]([x.sub.4])> = <8, ((0.2,0.3,0.4), (0.4,0.5,0.6))>,

that we desire to aggregate using the weighting vector w =(0.2,0.4,0.l,0.3). Performing the ordering the FNIFOWG pairs with respect to the first compoent, we get:

<[u.sub.[sigma](1)], [??]([x.sub.[sigma](1)])> = <15,((0.4,0.5,0.5),(0.1,0.1,0.2))>, <[u.sub.[sigma](2)], [??]([x.sub.[sigma](2)])> = <12,((0.3,0.4,0.5),(0.1,0.2,0.3))>, <[u.sub.[sigma](3)], [??]([x.sub.[sigma](3)])> = <10,((0.2,0.3,0.3),(0.3,0.4,0.5))>, <[u.sub.[sigma](4)], [??]([x.sub.[sigma](4)])> = <8((0.2,0.3,0.4),(0.4,0.5,0.6))>.

This ordering includes the ordered fuzzy number intuitionistic fuzzy numbers:

[??]([x.sub.[sigma](1)]) = <(0.4,0.5,0.5), (0.1,0.1,0.2)>, [??]([x.sub.[sigma](2)]) = <(0.3,0.4,0.5), (0.1,0.2,0.3)>, [??]([x.sub.[sigma](3)]) = <(0.2,0.3,0.3), (0.3,0.4,0.5)>, [??]([x.sub.[sigma](4)]) = <(0.2,0.3,0.4), (0.4,0.5,0.6)>.

Suppose the fuzzy measure of attribute of [[??].sub.i] (i = 1,2,3,4) and attribute sets of [[??].sub.i] (i=l,2,3,4) as follows:

[mu]([phi]) = 0, [mu]([[??].sub.1]) = 0.40, [mu]([[??].sub.2]) = 0.25, [mu]([[??].sub.3]) = 0.38, [mu]([[??].sub.4]) = 0.25, [mu]([[??].sub.1], [[??].sub.2]) = 0.56, [mu]([[??].sub.1], [[??].sub.3]) = 0.65, [mu]([[??].sub.1], [[??].sub.4]) = 0.50, [mu]([[??].sub.2], [[??].sub.3]) = 0.45, [mu]([[??].sub.2], [[??].sub.4]) = 0.39, [mu]([[??].sub.3], [[??].sub.4]) = 0.40, [mu]([[??].sub.1], [[??].sub.2], [[??].sub.3]) = 0.80, [mu]([[??].sub.1], [[??].sub.2], [[??].sub.4]) = 0.75, [mu]([[??].sub.1], [[??].sub.2], [[??].sub.4]) = 0.72, [mu]([[??].sub.2], [[??].sub.3], [[??].sub.4]) = 0.60, [mu]([[??].sub.1], [[??].sub.2], [[??].sub.3], [[??].sub.4]) = 1.00.

Then, by (14), it follows that:

[IFNIFCOG.sub.[mu]] (<[u.sub.1], [??]([x.sub.1])>, <[u.sub.2], [??]([x.sub.2])>, <[u.sub.3], [??]([x.sub.3])>, <[u.sub.4], [??]([x.sub.4])>) = <([0.4.sup.0.25] x [0.3.sup.0.56-0.25] x [0.2.sup.0.80-0.56] x [0.1.sup.1-0.80], [0.5.sup.0.25] x [0.4.sup.0.56-0.25] x [0.3.sup.0.80-0.56] x [0.2.sup.1-0.80], [0.5.sup.0.25] x [0.5.sup.0.56-0.25] x [0.3.sup.0.80-0.56] x [0.3.sup.1-0.80]), (1 - [(1 - 0.1).sup.0.25] x [(1 - 0.1).sup.0.56-0.25] x [(1 - 0.3).sup.0.80- 0.56] x [(1 - 0.3).sup.1-0.80] J 1 - [(1 - 0.1).sup.0.25] x [(1 - 0.2).sup.0.56-0.25] x [(1 - 0.4).sup.0.80-0.56] x [(1 - 0.4).sup.1-0.80] J 1 - [(1 - 0.2).sup.0.25] x [(1 - 0.3).sup.0.56-0.25] x [(1 - 0.5).sup.0.80-0.56] x [(1 - 0.5).sup.1-0.80])> = <(0.292,0.344,0.399), (0.194,0.274,0.661)>.

3. An approach to multiple attribute decision making with fuzzy number intuitionistic fuzzy information

In this section, we shall develop an approach to multiple attribute decision making with fuzzy number intuitionistic fuzzy information as follows.

Let A = {[A.sub.1], [A.sub.2], ..., [A.sub.m]} be a discrete set of alternatives, and G = {[G.sub.1], [G.sub.2], ..., [G.sub.n]} be the set of attributes, [omega] = ([[omega].sub.1], [[omega].sub.1], ..., [[omega].sub.n]) is the weighting vector of the attribute [G.sub.j] (j = 1,2, ..., n), where [[omega].sub.j] [member of] [0,1], [n.summation over (j=1)] [[omega].sub.j] = 1. Suppose that [??] = [[([[??].sub.ij]).sub.mxn] = <([a.sub.ij], [b.sub.ij], [c.sub.ij]),([l.sub.ij], [m.sub.ij], [p.sub.ij])>.sub.mxn] is the fuzzy number intuitionistic fuzzy decision matrix, where ([a.sub.ij], [b.sub.ij], [c.sub.ij]) indicates the degree that the alternative [A.sub.i] satisfies the attribute [G.sub.j], ([l.sub.ij], [m.sub.ij], [p.sub.ij]) indicates the degree that the alternative [A.sub.i] doesn't satisfy the attribute [G.sub.j] ([a.sub.ij], [b.sub.ij], [c.sub.ij]) [subset] [0,l], ([l.sub.ij], [m.sub.ij], [p.sub.ij]) [subset] [0,l], [c.sub.ij] + [p.sub.ij] [less than or equal to] 1, i = 1,2 , ..., m, j = 1, 2, ..., n.

In the following, we apply the IFNIFCOA and IFNIFCOG operator to multiple attribute decision making with fuzzy number intuitionistic fuzzy information.

Step 1. Calculate the order-inducing variables [([u.sub.ij]).sub.mxn] to be used in the decision matrix for each alternative [A.sub.i] and the attribute [G.sub.j]. The experts use order-inducing variables to represent the complex attitudinal character involving the opinion of different members of the board of directors (Merigo, Gil-Lafuente 2009).

Step 2. If we emphasize the group's influence, utilize the decision information given in matrix [??], and the IFNIFCOA operator:

[[??].sub.i] = <([a.sub.j], [b.sub.j], [c.sub.j]), ([l.sub.j], [m.sub.j], [p.sub.j])) =

IFNIFCOA (<[u.sub.i1], [??]([x.sub.i1])>, (<[u.sub.i2], [??]([x.sub.i2])>, ..., (<[u.sub.in], [??]([x.sub.in])>),

i = 1,2, ..., m, j = 1, 2, ..., n (15)

to derive the overall preference values [[??].sub.i](i = 1,2, ..., m) of the alternative [A.sub.i]. Otherwise utilize the decision information given in matrix [??], and the IFNIFCOG operator:

[[??].sub.i] = <([a.sub.j], [b.sub.j], [c.sub.j]), ([l.sub.j], [m.sub.j], [p.sub.j])) =

IFNIFCOG (<[u.sub.i1], [??]([x.sub.i1])>, (<[u.sub.i2], [??]([x.sub.i2])>, ..., (<[u.sub.in], [??]([x.sub.in])>),

i = 1,2, ..., m, j = 1, 2, ..., n (16)

to derive the overall preference values [[??].sub.i](i = 1,2, ...,m) of the alternative [A.sub.i].

Step 3. Calculate the scores S ([[??].sub.i]) (i = 1,2, ..., m) of the overall fuzzy number intuitionistic fuzzy preference values [[??].sub.i] (i = 1,2, ..., m) to rank all the alternatives [[??].sub.i] (i = 1,2, ..., m) and then to select the best one(s) (if there is no difference between two scores S ([[??].sub.i]) and S ([[??].sub.j]), then we need to calculate the accuracy degrees H ([[??].sub.i]) and H ([[??].sub.j]) of the overall fuzzy number intuitionistic fuzzy preference values r i and ry , respectively, and then rank the alternatives [A.sub.i] and [A.sub.j] in accordance with the accuracy degrees H ([[??].sub.i]) and H ([[??].sub.i]).

Step 4. Rank all the alternatives [A.sub.i] (i = 1,2, ..., m) and select the best one(s) in accordance with S([[??].sub.i]) and H([[??].sub.i]) (i = 1,2, ..., m).

Step 5. End.

4. Illustrative example

In this section we shall present a numerical example to show potential evaluation of emerging technology commercialization with fuzzy number intuitionistic fuzzy information in order to illustrate the method proposed in this paper. There is a panel with five possible emerging technology enterprises [A.sub.1] (i = 1,2,3,4,5) to select. The experts selects four attribute to evaluate the four possible emerging technology enterprises: (1)G1 is the technical advancement; (2)G2 is the potential market and market risk; (3)G3 is the industrialization infrastructure, human resources and financial conditions; (4)G4 is the employment creation and the development of science and technology. The five possible enterprises A' (i = 1,2,3,4,5) are to be evaluated using using the fuzzy number intuitionistic fuzzy numbers by the decision maker under the above four attributes, and construct, respectively, the decision matrices as listed in the following matrices

[[??].sub.k] = [([[??].sup.(k).sub.ij]).sub.5x4] (k = 1,2,3) as follows:

[??] =

[<(0.2,0.3,0.4), (0.3,0.4,0.4)> <(0.5,0.6,0.7), (0.1,0.1,0.1)>

<(0.3,0.4,0.5), (0.1,0.2,0.3)> <(0.4,0.5,0.5), (0.1,0.1,0.2)>

<(0.1,0.2,0.3), (0.3,0.4,0.5)> <(0.3,0.4,0.5), (0.2,0.2,0.3)>

<(0.4,0.5,0.6), (0.1,0.1,0.1)> <(0.7,0.7,0.7), (0.1,0.1,0.1)>

<(0.6,0.6,0.7), (0.1,0.1,0.1)> <(0.4,0.4,0.4), (0.1,0.2,0.3)>

<(0.5,0.5,0.6), (0.1,0.1,0.2)> <(0.4,0.5,0.6), (0.1,0.1,0.1)>

<(0.3,0.4,0.5), (0.1,0.2,0.3)> <(0.1,0.2,0.3), (0.3,0.4,0.5)>

<(0.6,0.7,0.8), (0.1,0.1,0.1)> <(0.1,0.1,0.2), (0.4,0.5,0.6)> .

<(0.4,0.5,0.5), (0.1,0.2,0.3)> <(0.3,0.4,0.5), (0.2,0.3,0.3)>

<(0.6,0.6,0.6), (0.1,0.1,0.1)> <(0.2,0.3,0.3), (0.3,0.4,0.5)>]

Then, we utilize the approach developed to get the most desirable alternative(s).

Step 1. Suppose the fuzzy measure of attribute of [G.sub.j] (j = 1,2, ..., n) and attribute sets of G as follows:

[mu]([G.sub.1]) = 0.30, [mu]([G.sub.2]) = 0.35, [mu]([G.sub.3]) = 0.30, [mu]([G.sub.4]) = 0.22, [mu]([G.sub.1], [G.sub.2]) = 0.70, [mu]([G.sub.1], [G.sub.3]) = 0.60, [mu]([G.sub.1], [G.sub.4]) = 0.55, [mu]([G.sub.2], [G.sub.3]) = 0.50, [mu]([G.sub.2], [G.sub.4]) = 0.45, [mu]([G.sub.3], [G.sub.4]) = 0.40, [mu]([G.sub.1], [G.sub.2], [G.sub.3]) = 0.82, [mu]([G.sub.1], [G.sub.2], [G.sub.4]) = 0.87,[mu]([G.sub.1], [G.sub.3], [G.sub.4]) = 0.75, [mu]([G.sub.2], [G.sub.3], [G.sub.4]) = 0.60, [mu]([G.sub.1], [G.sub.2], [G.sub.3], [G.sub.4]) = 1.00.

Step 2. The experts use order-inducing variables to represent the complex attitudinal character involving the opinion of different members of the board of directors (Merigo, Gil-Lafuente 2009). The results are shown in Table 1.

Step 3. If we emphasize the group's influence, we utilize the decision information given in matrix [??], and the IFNIFCOA operator to obtain the overall preference values [[??].sub.i] of the alternatives [A.sub.i] (i = 1, 2, ..., 5).

[[??].sub.1] = <(0.385,0.471,0.575), (0.155,0.174,0.193)>,

[[??].sub.2] = <(0.306,0.407,0.469), (0.122,0.178,0.285)>,

[[??].sub.3] = <(0.329,0.425,0.540), (0.210,0.239,0.288)>,

[[??].sub.4] = <(0.516,0.568,0.613), (0.113,0.132,0.139)>,

[[??].sub.5] = <(0.505,0.516,0.556), (0.122,0.149,0.170)>.

Step 4. Calculate the scores S([[??].sub.i]) (i = 1,2, ..., 5)of the overall fuzzy number intuitionistic fuzzy preference values [[??].sub.i] (i = 1,2, ..., 5):

S([[??].sub.i]) = 0.603, S([[??].sub.2]) = 0.413, S([[??].sub.3]) = 0.372, S([[??].sub.4]) = 0.874, S([[??].sub.5]) = 0.751.

Step 5. Rank all the alternatives [A.sub.i] (i = 1,2,3,4,5) in accordance with the scores S ([??].sub.i]) (i = 1, 2, ..., 5) of the overall fuzzy number intuitionistic fuzzy preference values [[??].sub.i] (i = 1,2, ..., 5) : [A.sub.4] [??] [A.sub.5] [??] [A.sub.1] [??] [A.sub.2] [??] [A.sub.3], and thus the most desirable alternative is [A.sub.4]. If we emphasize the individual influence, we utilize the decision information given in matrix [R] and the IFNIFCOG operator to obtain the overall preference values [[??].sub.i] of the alternatives [A.sub.i] (i = 1,2, ..., 5).

[[??].sub.1] = <(0.339,0.434,0.538), (0.186,0.235,0.248)>,

[[??].sub.2] = <(0.272,0.382,0.456), (0.140,0.208,0.310)>,

[[??].sub.3] = <(0.213,0.295,0.415), (0.246,0.305,0.387)>,

[[??].sub.4] = <(0.462,0.540,0.600), (0.119,0.152,0.165)>,

[[??].sub.5] = <(0.450,0.484,0.507), (0.140,0.185,0.234)>.

Then, by applying Eqs (4) to calculate the scores S([[??].sub.i]) (i = 1,2, ...,5) of the collective overall fuzzy number intuitionistic fuzzy preference values [[??].sub.i] (i = 1,2, ..., 5):

S ([[??].sub.1]) = 0.421, S ([[??].sub.2]) = 0.313, S ([[??].sub.3]) = -0.013, s ([[??].sub.4]) = 0.777, s ([[??].sub.5]) = 0.592.

Therefore, the ranking order is [A.sub.4] [??] [A.sub.5] [??] [A.sub.1] [??] [A.sub.2] [??] [A.sub.3]. Thus, we can see that the most desirable alternative is still [A.sub.4].

Especially, if the triangular fuzzy numbers ([a.sub.j], [b.sub.j], [c.sub.j]) and ([l.sub.j], [m.sub.j], [p.sub.j]) are reduced to the interval numbers [ay, by ] and [ly, my ], then, the IFNIFCOA or IFNIFCOG operator is reduced to the induced interval-valued intuitionistic fuzzy choquet ordered averaging (I- IVIFCOA) operator or induced interval-valued intuitionistic fuzzy choquet ordered geometric (I-IVIFCOG) operator (Xu, Xia 2011); if [a.sub.j] = [b.sub.j] = [c.sub.j] = [[mu].sub.j], [l.sub.j] = [m.sub.j] = [p.sub.j] = [v.sub.j], then the IFNIFCOA or IFNIFCOG operator is reduced to the induced intuitionistic fuzzy choquet ordered averaging (I-IFCOA) operator or induced intuitionistic fuzzy choquet ordered geometric (I-IFCOG) operator (Xu, Xia 2011).

Conclusions

The traditional induced Choquet integral aggregation operators are generally suitable for aggregating the information taking the form of numerical values, and yet they will fail in dealing with fuzzy number intuitionistic fuzzy information. In this paper, we have developed two induced fuzzy number intuitionistic fuzzy Choquet integral aggregation operators: induced fuzzy number intuitionistic fuzzy choquet ordered averaging (IFNIFCOA) operator and induced fuzzy number intuitionistic fuzzy choquet ordered geometric (IFNIFCOG) operator. The prominent characteristic of the operators is that they can not only consider the importance of the elements or their ordered positions, but also reflect the correlation among the elements or their ordered positions. We have studied some desirable properties of the IFNIFCOA and IFNIFCOG operators, such as commutativity, idempotency and monotonicity, and applied the IFNIFCOA and IFNIFCOGM operators to multiple attribute decision making with fuzzy number intuitionistic fuzzy information. Finally an illustrative example has been given to show the developed method. In the future, we shall continue working in the application of the fuzzy number intuitionistic fuzzy multiple attribute decision making to other domains.

Acknowledgment

The author is very grateful to the editor and the anonymous referees for their insightful and constructive comments and suggestions, which have been very helpful in improving the paper. The work was supported by the National Natural Science Foundation of China under Grant No. 61174149, Natural Science Foundation Project of CQ CSTC of the People's Republic of China (No. CSTC, 2011BA0035) and the Humanities and Social Sciences Foundation of Ministry of Education of the People's Republic of China under Grant No.11XJC630011.

doi: 10.3846/16111699.2012.707984

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Guiwu Wei [1], Rui Lin [2], Xiaofei Zhao [3], Hongjun Wang [4]

School of Economics and Management, Chongqing University of Arts and Sciences, Chongqing 402160, P. R. China

E-mails: [1] weiguiwu@163.com (corresponding author); [2] linrui20000@163.com; [3] zhao_xiaofei1980@163.com; [4] wang_hongjun2008@163.com

Received 09 November 2011; accepted 27 June 2012

Guiwu WEI has a MSc and a PhD degree in applied mathematics from SouthWest Petroleum University, Business Administration from School of Economics and Management at SouthWest Jiaotong University, China, respectively. He is an Associate Professor in the Department of Economics and Management at Chongqing University of Arts and Sciences. He has published more than 90 papers in journals, books and conference proceedings including journals such as Expert Systems with Applications, Applied Soft Computing, Knowledge and Information Systems, Knowledge-based Systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, International Journal of Computational Intelligence Systems and Information: An International Journal. He has published 1 book. He has participated in several scientific committees and serves as a reviewer in a wide range of journals including Computers & Industrial Engineering, International Journal of Information Technology and Decision Making, Knowledge-based Systems, Information Sciences, International Journal of Computational Intelligence Systems and European Journal of Operational Research. He is currently interested in Aggregation Operators, Decision Making and Computing with Words.

Rui LIN is a lecturer in Department of Economics and Management, Chongqing University of Arts and Sciences. He received the B.E. degree in management sciences and engineer from Chengdu University of Technology, China. He has worked for Department of Economics and Management, Chongqing University of Arts and Sciences, China as a lecturer since 2007.

Xiaofei ZHAO is a lecturer in Department of Economics and Management, Chongqing University of Arts and Sciences. He received the B.E. degree in management sciences and engineer from SouthWest Jiaotong University, China. He has worked for Department of Economics and Management, Chongqing University of Arts and Sciences, China as a lecturer since 2010.

Hongjun WANG is a lecturer in Department of Economics and Management, Chongqing University of Arts and Sciences. She received the B.E.and M.E. degree in management sciences and engineer from SouthWest Petroleum University, China. He has worked for Department of Economics and Management, Chongqing University of Arts and Sciences, China as a lecturer since 2006.

Table 1. Inducing variables

            [S.sub.1]   [S.sub.2]   [S.sub.3]   [S.sub.4]

[A.sub.1]   12          18          16          15
[A.sub.2]   23          25          22          20
[A.sub.3]   18          20          24          14
[A.sub.4]   22          24          20          18
[A.sub.5]   24          18          21          16